Eigen/src/Geometry/Hyperplane.h - eigen - Git at Google

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
 // Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
 //
 // This Source Code Form is subject to the terms of the Mozilla
 // Public License v. 2.0. If a copy of the MPL was not distributed
 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

 #ifndef EIGEN_HYPERPLANE_H
 #define EIGEN_HYPERPLANE_H

 // IWYU pragma: private
 #include "./InternalHeaderCheck.h"

 namespace Eigen {

 /** \geometry_module \ingroup Geometry_Module
  *
  * \class Hyperplane
  *
  * \brief A hyperplane
  *
  * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n.
  * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane.
  *
  * \tparam Scalar_ the scalar type, i.e., the type of the coefficients
  * \tparam AmbientDim_ the dimension of the ambient space, can be a compile time value or Dynamic.
  *             Notice that the dimension of the hyperplane is AmbientDim_-1.
  *
  * This class represents an hyperplane as the zero set of the implicit equation
  * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part)
  * and \f$ d \f$ is the distance (offset) to the origin.
  */
 template <typename Scalar_, int AmbientDim_, int Options_>
 class Hyperplane {
  public:
   EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_,
                                                              AmbientDim_ == Dynamic ? Dynamic : AmbientDim_ + 1)
   enum { AmbientDimAtCompileTime = AmbientDim_, Options = Options_ };
   typedef Scalar_ Scalar;
   typedef typename NumTraits<Scalar>::Real RealScalar;
   typedef Eigen::Index Index;  ///< \deprecated since Eigen 3.3
   typedef Matrix<Scalar, AmbientDimAtCompileTime, 1> VectorType;
   typedef Matrix<Scalar, Index(AmbientDimAtCompileTime) == Dynamic ? Dynamic : Index(AmbientDimAtCompileTime) + 1, 1,
                  Options>
       Coefficients;
   typedef Block<Coefficients, AmbientDimAtCompileTime, 1> NormalReturnType;
   typedef const Block<const Coefficients, AmbientDimAtCompileTime, 1> ConstNormalReturnType;

   /** Default constructor without initialization */
   EIGEN_DEVICE_FUNC inline Hyperplane() {}

   template <int OtherOptions>
   EIGEN_DEVICE_FUNC Hyperplane(const Hyperplane<Scalar, AmbientDimAtCompileTime, OtherOptions>& other)
       : m_coeffs(other.coeffs()) {}

   /** Constructs a dynamic-size hyperplane with \a _dim the dimension
    * of the ambient space */
   EIGEN_DEVICE_FUNC inline explicit Hyperplane(Index _dim) : m_coeffs(_dim + 1) {}

   /** Construct a plane from its normal \a n and a point \a e onto the plane.
    * \warning the vector normal is assumed to be normalized.
    */
   EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const VectorType& e) : m_coeffs(n.size() + 1) {
     normal() = n;
     offset() = -n.dot(e);
   }

   /** Constructs a plane from its normal \a n and distance to the origin \a d
    * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$.
    * \warning the vector normal is assumed to be normalized.
    */
   EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const Scalar& d) : m_coeffs(n.size() + 1) {
     normal() = n;
     offset() = d;
   }

   /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space
    * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made.
    */
   EIGEN_DEVICE_FUNC static inline Hyperplane Through(const VectorType& p0, const VectorType& p1) {
     Hyperplane result(p0.size());
     result.normal() = (p1 - p0).unitOrthogonal();
     result.offset() = -p0.dot(result.normal());
     return result;
   }

   /** Constructs a hyperplane passing through the three points. The dimension of the ambient space
    * is required to be exactly 3.
    */
   EIGEN_DEVICE_FUNC static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2) {
     EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3)
     Hyperplane result(p0.size());
     VectorType v0(p2 - p0), v1(p1 - p0);
     result.normal() = v0.cross(v1);
     RealScalar norm = result.normal().norm();
     if (norm <= v0.norm() * v1.norm() * NumTraits<RealScalar>::epsilon()) {
       Matrix<Scalar, 2, 3> m;
       m << v0.transpose(), v1.transpose();
       JacobiSVD<Matrix<Scalar, 2, 3>, ComputeFullV> svd(m);
       result.normal() = svd.matrixV().col(2);
     } else
       result.normal() /= norm;
     result.offset() = -p0.dot(result.normal());
     return result;
   }

   /** Constructs a hyperplane passing through the parametrized line \a parametrized.
    * If the dimension of the ambient space is greater than 2, then there isn't uniqueness,
    * so an arbitrary choice is made.
    */
   // FIXME to be consistent with the rest this could be implemented as a static Through function ??
   EIGEN_DEVICE_FUNC explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized) {
     normal() = parametrized.direction().unitOrthogonal();
     offset() = -parametrized.origin().dot(normal());
   }

   EIGEN_DEVICE_FUNC ~Hyperplane() {}

   /** \returns the dimension in which the plane holds */
   EIGEN_DEVICE_FUNC inline Index dim() const {
     return AmbientDimAtCompileTime == Dynamic ? m_coeffs.size() - 1 : Index(AmbientDimAtCompileTime);
   }

   /** normalizes \c *this */
   EIGEN_DEVICE_FUNC void normalize(void) { m_coeffs /= normal().norm(); }

   /** \returns the signed distance between the plane \c *this and a point \a p.
    * \sa absDistance()
    */
   EIGEN_DEVICE_FUNC inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); }

   /** \returns the absolute distance between the plane \c *this and a point \a p.
    * \sa signedDistance()
    */
   EIGEN_DEVICE_FUNC inline Scalar absDistance(const VectorType& p) const { return numext::abs(signedDistance(p)); }

   /** \returns the projection of a point \a p onto the plane \c *this.
    */
   EIGEN_DEVICE_FUNC inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); }

   /** \returns a constant reference to the unit normal vector of the plane, which corresponds
    * to the linear part of the implicit equation.
    */
   EIGEN_DEVICE_FUNC inline ConstNormalReturnType normal() const {
     return ConstNormalReturnType(m_coeffs, 0, 0, dim(), 1);
   }

   /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds
    * to the linear part of the implicit equation.
    */
   EIGEN_DEVICE_FUNC inline NormalReturnType normal() { return NormalReturnType(m_coeffs, 0, 0, dim(), 1); }

   /** \returns the distance to the origin, which is also the "constant term" of the implicit equation
    * \warning the vector normal is assumed to be normalized.
    */
   EIGEN_DEVICE_FUNC inline const Scalar& offset() const { return m_coeffs.coeff(dim()); }

   /** \returns a non-constant reference to the distance to the origin, which is also the constant part
    * of the implicit equation */
   EIGEN_DEVICE_FUNC inline Scalar& offset() { return m_coeffs(dim()); }

   /** \returns a constant reference to the coefficients c_i of the plane equation:
    * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
    */
   EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }

   /** \returns a non-constant reference to the coefficients c_i of the plane equation:
    * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
    */
   EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }

   /** \returns the intersection of *this with \a other.
    *
    * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines.
    *
    * \note If \a other is approximately parallel to *this, this method will return any point on *this.
    */
   EIGEN_DEVICE_FUNC VectorType intersection(const Hyperplane& other) const {
     EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
     Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0);
     // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests
     // whether the two lines are approximately parallel.
     if (internal::isMuchSmallerThan(det, Scalar(1))) {  // special case where the two lines are approximately parallel.
                                                         // Pick any point on the first line.
       if (numext::abs(coeffs().coeff(1)) > numext::abs(coeffs().coeff(0)))
         return VectorType(coeffs().coeff(1), -coeffs().coeff(2) / coeffs().coeff(1) - coeffs().coeff(0));
       else
         return VectorType(-coeffs().coeff(2) / coeffs().coeff(0) - coeffs().coeff(1), coeffs().coeff(0));
     } else {  // general case
       Scalar invdet = Scalar(1) / det;
       return VectorType(
           invdet * (coeffs().coeff(1) * other.coeffs().coeff(2) - other.coeffs().coeff(1) * coeffs().coeff(2)),
           invdet * (other.coeffs().coeff(0) * coeffs().coeff(2) - coeffs().coeff(0) * other.coeffs().coeff(2)));
     }
   }

   /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this.
    *
    * \param mat the Dim x Dim transformation matrix
    * \param traits specifies whether the matrix \a mat represents an #Isometry
    *               or a more generic #Affine transformation. The default is #Affine.
    */
   template <typename XprType>
   EIGEN_DEVICE_FUNC inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine) {
     if (traits == Affine) {
       normal() = mat.inverse().transpose() * normal();
       m_coeffs /= normal().norm();
     } else if (traits == Isometry)
       normal() = mat * normal();
     else {
       eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
     }
     return *this;
   }

   /** Applies the transformation \a t to \c *this and returns a reference to \c *this.
    *
    * \param t the transformation of dimension Dim
    * \param traits specifies whether the transformation \a t represents an #Isometry
    *               or a more generic #Affine transformation. The default is #Affine.
    *               Other kind of transformations are not supported.
    */
   template <int TrOptions>
   EIGEN_DEVICE_FUNC inline Hyperplane& transform(const Transform<Scalar, AmbientDimAtCompileTime, Affine, TrOptions>& t,
                                                  TransformTraits traits = Affine) {
     transform(t.linear(), traits);
     offset() -= normal().dot(t.translation());
     return *this;
   }

   /** \returns \c *this with scalar type casted to \a NewScalarType
    *
    * Note that if \a NewScalarType is equal to the current scalar type of \c *this
    * then this function smartly returns a const reference to \c *this.
    */
   template <typename NewScalarType>
   EIGEN_DEVICE_FUNC inline
       typename internal::cast_return_type<Hyperplane,
                                           Hyperplane<NewScalarType, AmbientDimAtCompileTime, Options> >::type
       cast() const {
     return
         typename internal::cast_return_type<Hyperplane,
                                             Hyperplane<NewScalarType, AmbientDimAtCompileTime, Options> >::type(*this);
   }

   /** Copy constructor with scalar type conversion */
   template <typename OtherScalarType, int OtherOptions>
   EIGEN_DEVICE_FUNC inline explicit Hyperplane(
       const Hyperplane<OtherScalarType, AmbientDimAtCompileTime, OtherOptions>& other) {
     m_coeffs = other.coeffs().template cast<Scalar>();
   }

   /** \returns \c true if \c *this is approximately equal to \a other, within the precision
    * determined by \a prec.
    *
    * \sa MatrixBase::isApprox() */
   template <int OtherOptions>
   EIGEN_DEVICE_FUNC bool isApprox(
       const Hyperplane<Scalar, AmbientDimAtCompileTime, OtherOptions>& other,
       const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const {
     return m_coeffs.isApprox(other.m_coeffs, prec);
   }

  protected:
   Coefficients m_coeffs;
 };

 }  // end namespace Eigen

 #endif  // EIGEN_HYPERPLANE_H
	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
	// Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
	//
	// This Source Code Form is subject to the terms of the Mozilla
	// Public License v. 2.0. If a copy of the MPL was not distributed
	// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

	#ifndef EIGEN_HYPERPLANE_H
	#define EIGEN_HYPERPLANE_H

	// IWYU pragma: private
	#include "./InternalHeaderCheck.h"

	namespace Eigen {

	/** \geometry_module \ingroup Geometry_Module
	*
	* \class Hyperplane
	*
	* \brief A hyperplane
	*
	* A hyperplane is an affine subspace of dimension n-1 in a space of dimension n.
	* For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane.
	*
	* \tparam Scalar_ the scalar type, i.e., the type of the coefficients
	* \tparam AmbientDim_ the dimension of the ambient space, can be a compile time value or Dynamic.
	* Notice that the dimension of the hyperplane is AmbientDim_-1.
	*
	* This class represents an hyperplane as the zero set of the implicit equation
	* \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part)
	* and \f$ d \f$ is the distance (offset) to the origin.
	*/
	template <typename Scalar_, int AmbientDim_, int Options_>
	class Hyperplane {
	public:
	EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_,
	AmbientDim_ == Dynamic ? Dynamic : AmbientDim_ + 1)
	enum { AmbientDimAtCompileTime = AmbientDim_, Options = Options_ };
	typedef Scalar_ Scalar;
	typedef typename NumTraits<Scalar>::Real RealScalar;
	typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
	typedef Matrix<Scalar, AmbientDimAtCompileTime, 1> VectorType;
	typedef Matrix<Scalar, Index(AmbientDimAtCompileTime) == Dynamic ? Dynamic : Index(AmbientDimAtCompileTime) + 1, 1,
	Options>
	Coefficients;
	typedef Block<Coefficients, AmbientDimAtCompileTime, 1> NormalReturnType;
	typedef const Block<const Coefficients, AmbientDimAtCompileTime, 1> ConstNormalReturnType;

	/** Default constructor without initialization */
	EIGEN_DEVICE_FUNC inline Hyperplane() {}

	template <int OtherOptions>
	EIGEN_DEVICE_FUNC Hyperplane(const Hyperplane<Scalar, AmbientDimAtCompileTime, OtherOptions>& other)
	: m_coeffs(other.coeffs()) {}

	/** Constructs a dynamic-size hyperplane with \a _dim the dimension
	* of the ambient space */
	EIGEN_DEVICE_FUNC inline explicit Hyperplane(Index _dim) : m_coeffs(_dim + 1) {}

	/** Construct a plane from its normal \a n and a point \a e onto the plane.
	* \warning the vector normal is assumed to be normalized.
	*/
	EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const VectorType& e) : m_coeffs(n.size() + 1) {
	normal() = n;
	offset() = -n.dot(e);
	}

	/** Constructs a plane from its normal \a n and distance to the origin \a d
	* such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$.
	* \warning the vector normal is assumed to be normalized.
	*/
	EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const Scalar& d) : m_coeffs(n.size() + 1) {
	normal() = n;
	offset() = d;
	}

	/** Constructs a hyperplane passing through the two points. If the dimension of the ambient space
	* is greater than 2, then there isn't uniqueness, so an arbitrary choice is made.
	*/
	EIGEN_DEVICE_FUNC static inline Hyperplane Through(const VectorType& p0, const VectorType& p1) {
	Hyperplane result(p0.size());
	result.normal() = (p1 - p0).unitOrthogonal();
	result.offset() = -p0.dot(result.normal());
	return result;
	}

	/** Constructs a hyperplane passing through the three points. The dimension of the ambient space
	* is required to be exactly 3.
	*/
	EIGEN_DEVICE_FUNC static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2) {
	EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3)
	Hyperplane result(p0.size());
	VectorType v0(p2 - p0), v1(p1 - p0);
	result.normal() = v0.cross(v1);
	RealScalar norm = result.normal().norm();
	if (norm <= v0.norm() * v1.norm() * NumTraits<RealScalar>::epsilon()) {
	Matrix<Scalar, 2, 3> m;
	m << v0.transpose(), v1.transpose();
	JacobiSVD<Matrix<Scalar, 2, 3>, ComputeFullV> svd(m);
	result.normal() = svd.matrixV().col(2);
	} else
	result.normal() /= norm;
	result.offset() = -p0.dot(result.normal());
	return result;
	}

	/** Constructs a hyperplane passing through the parametrized line \a parametrized.
	* If the dimension of the ambient space is greater than 2, then there isn't uniqueness,
	* so an arbitrary choice is made.
	*/
	// FIXME to be consistent with the rest this could be implemented as a static Through function ??
	EIGEN_DEVICE_FUNC explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized) {
	normal() = parametrized.direction().unitOrthogonal();
	offset() = -parametrized.origin().dot(normal());
	}

	EIGEN_DEVICE_FUNC ~Hyperplane() {}

	/** \returns the dimension in which the plane holds */
	EIGEN_DEVICE_FUNC inline Index dim() const {
	return AmbientDimAtCompileTime == Dynamic ? m_coeffs.size() - 1 : Index(AmbientDimAtCompileTime);
	}

	/** normalizes \c this /
	EIGEN_DEVICE_FUNC void normalize(void) { m_coeffs /= normal().norm(); }

	/** \returns the signed distance between the plane \c *this and a point \a p.
	* \sa absDistance()
	*/
	EIGEN_DEVICE_FUNC inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); }

	/** \returns the absolute distance between the plane \c *this and a point \a p.
	* \sa signedDistance()
	*/
	EIGEN_DEVICE_FUNC inline Scalar absDistance(const VectorType& p) const { return numext::abs(signedDistance(p)); }

	/** \returns the projection of a point \a p onto the plane \c *this.
	*/
	EIGEN_DEVICE_FUNC inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); }

	/** \returns a constant reference to the unit normal vector of the plane, which corresponds
	* to the linear part of the implicit equation.
	*/
	EIGEN_DEVICE_FUNC inline ConstNormalReturnType normal() const {
	return ConstNormalReturnType(m_coeffs, 0, 0, dim(), 1);
	}

	/** \returns a non-constant reference to the unit normal vector of the plane, which corresponds
	* to the linear part of the implicit equation.
	*/
	EIGEN_DEVICE_FUNC inline NormalReturnType normal() { return NormalReturnType(m_coeffs, 0, 0, dim(), 1); }

	/** \returns the distance to the origin, which is also the "constant term" of the implicit equation
	* \warning the vector normal is assumed to be normalized.
	*/
	EIGEN_DEVICE_FUNC inline const Scalar& offset() const { return m_coeffs.coeff(dim()); }

	/** \returns a non-constant reference to the distance to the origin, which is also the constant part
	* of the implicit equation */
	EIGEN_DEVICE_FUNC inline Scalar& offset() { return m_coeffs(dim()); }

	/** \returns a constant reference to the coefficients c_i of the plane equation:
	* \f$ c_0x_0 + ... + c_{d-1}x_{d-1} + c_d = 0 \f$
	*/
	EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }

	/** \returns a non-constant reference to the coefficients c_i of the plane equation:
	* \f$ c_0x_0 + ... + c_{d-1}x_{d-1} + c_d = 0 \f$
	*/
	EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }

	/** \returns the intersection of *this with \a other.
	*
	* \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines.
	*
	* \note If \a other is approximately parallel to this, this method will return any point on this.
	*/
	EIGEN_DEVICE_FUNC VectorType intersection(const Hyperplane& other) const {
	EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
	Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0);
	// since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests
	// whether the two lines are approximately parallel.
	if (internal::isMuchSmallerThan(det, Scalar(1))) { // special case where the two lines are approximately parallel.
	// Pick any point on the first line.
	if (numext::abs(coeffs().coeff(1)) > numext::abs(coeffs().coeff(0)))
	return VectorType(coeffs().coeff(1), -coeffs().coeff(2) / coeffs().coeff(1) - coeffs().coeff(0));
	else
	return VectorType(-coeffs().coeff(2) / coeffs().coeff(0) - coeffs().coeff(1), coeffs().coeff(0));
	} else { // general case
	Scalar invdet = Scalar(1) / det;
	return VectorType(
	invdet * (coeffs().coeff(1) * other.coeffs().coeff(2) - other.coeffs().coeff(1) * coeffs().coeff(2)),
	invdet * (other.coeffs().coeff(0) * coeffs().coeff(2) - coeffs().coeff(0) * other.coeffs().coeff(2)));
	}
	}

	/** Applies the transformation matrix \a mat to \c this and returns a reference to \c this.
	*
	* \param mat the Dim x Dim transformation matrix
	* \param traits specifies whether the matrix \a mat represents an #Isometry
	* or a more generic #Affine transformation. The default is #Affine.
	*/
	template <typename XprType>
	EIGEN_DEVICE_FUNC inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine) {
	if (traits == Affine) {
	normal() = mat.inverse().transpose() * normal();
	m_coeffs /= normal().norm();
	} else if (traits == Isometry)
	normal() = mat * normal();
	else {
	eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
	}
	return *this;
	}

	/** Applies the transformation \a t to \c this and returns a reference to \c this.
	*
	* \param t the transformation of dimension Dim
	* \param traits specifies whether the transformation \a t represents an #Isometry
	* or a more generic #Affine transformation. The default is #Affine.
	* Other kind of transformations are not supported.
	*/
	template <int TrOptions>
	EIGEN_DEVICE_FUNC inline Hyperplane& transform(const Transform<Scalar, AmbientDimAtCompileTime, Affine, TrOptions>& t,
	TransformTraits traits = Affine) {
	transform(t.linear(), traits);
	offset() -= normal().dot(t.translation());
	return *this;
	}

	/** \returns \c *this with scalar type casted to \a NewScalarType
	*
	* Note that if \a NewScalarType is equal to the current scalar type of \c *this
	* then this function smartly returns a const reference to \c *this.
	*/
	template <typename NewScalarType>
	EIGEN_DEVICE_FUNC inline
	typename internal::cast_return_type<Hyperplane,
	Hyperplane<NewScalarType, AmbientDimAtCompileTime, Options> >::type
	cast() const {
	return
	typename internal::cast_return_type<Hyperplane,
	Hyperplane<NewScalarType, AmbientDimAtCompileTime, Options> >::type(*this);
	}

	/** Copy constructor with scalar type conversion */
	template <typename OtherScalarType, int OtherOptions>
	EIGEN_DEVICE_FUNC inline explicit Hyperplane(
	const Hyperplane<OtherScalarType, AmbientDimAtCompileTime, OtherOptions>& other) {
	m_coeffs = other.coeffs().template cast<Scalar>();
	}

	/** \returns \c true if \c *this is approximately equal to \a other, within the precision
	* determined by \a prec.
	*
	* \sa MatrixBase::isApprox() */
	template <int OtherOptions>
	EIGEN_DEVICE_FUNC bool isApprox(
	const Hyperplane<Scalar, AmbientDimAtCompileTime, OtherOptions>& other,
	const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const {
	return m_coeffs.isApprox(other.m_coeffs, prec);
	}

	protected:
	Coefficients m_coeffs;
	};

	} // end namespace Eigen

	#endif // EIGEN_HYPERPLANE_H